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Assignment 1 Reading Assignment: 1. Chapter 1: Functions (From the course notes posted on piazza)2. Chapter 2: Metric Spaces and Topology (both from Moon’s book and course notes) Problems: 1. Let  x  = ( x 1 ,...,x n ) ,y  = ( y 1 ,...,y n ) ∈ R n and consider the function  ρ  given by ρ  x,y  = max {| x 1 − y 1 | ,..., | x n − y n |} . Show that  ρ  is a metric.2. Let  X   be a metric space with metric  d . Deﬁne ¯ d  :  X  × X   → R  by¯ d ( x,y ) = min { d ( x,y ) , 1 } . Show that ¯ d  is also a metric.3. The image of a function applied to a set-valued argument is deﬁned by  f  ( A )  { f  ( x ) | x ∈ A } and  f  − 1 ( B )  { x ∈ X  | f  ( x ) ∈ B } . Let  f   :  X   → Y   ,  A ⊆ X  , and  B  ⊆ Y   .(a) Show that  f  − 1 ( f  ( A )) ⊇ A  and that equality holds if   f   is injective.(b) Show that  f  ( f  − 1 ( B )) ⊆ B  and that equality holds if   f   is surjective.4. Let  ℓ 2 denote the subset of   R ω consisting of all sequences ( x 1 ,x 2 ,... ) such that   i  x 2 i  con-verges. (You may assume the standard facts about inﬁnite series.)(a) Show that if   x , y ∈ ℓ 2 , then   i | x i y i |  converges.(b) Let  c ∈ R . Show that if   x , y ∈ ℓ 2 , then so are  x + y  and  c x .(c) Show that d ( x , y ) =   ∞  i =1 ( x i − y i ) 2  1 / 2 is a well-deﬁne metric on  ℓ 2 . It is called the  ℓ 2 -metric.5. Let  x n  → x  and  y n  → y  in the space  R  with metric  d ( x,x ′ ) = | x − x ′ | . Show that x n  +  y n  → x  +  yx n − y n  → x − yx n y n  → xy, and provided that each  y n   = 0 and  y   = 0, x n /y n  → x/y. [Hint: First show that + , − , · ,/  are continuous functions from ( R 2 ,d 1 ) to ( R , |·| ).]1  6. Deﬁne  f  n  : [0 , 1] → R  by the equation  f  n ( x ) =  x n . Show that the sequence { f  n ( x ) } convergesfor each  x  ∈  [0 , 1], but that the sequence  { f  n }  does not converge uniformly. Recall thatuniform convergence to  f   on [ a,b ] implies that, for any  ǫ >  0, there exists an  N   such that | f  n ( t ) − f  ( t ) | < ǫ  for all  n > N   and all  t ∈ [ a,b ]. Optional Problems: 1. Show that  d ′ ( x,y ) =  d ( x,y ) / (1 +  d ( x,y )) is a bounded metric for  X   if   d  is a metric for  X  .2. Prove continuity of the algebraic operations on  R , as follows: Use the metric  d ( a,b ) = | a − b | on  R  and the metric ρ (( x,y ) , ( x 0 ,y 0 )) = max {| x − x 0 | , | y − y 0 |} on  R 2 .(a) Show that addition is continuous. [Hint: Given  ǫ , let  δ   =  ǫ/ 2 and note that d ( x  +  y,x 0  +  y 0 ) ≤| x − x 0 | + | y − y 0 | . ](b) Show that multiplication is continuous. [Hint: Given ( x 0 ,y 0 ) and  ǫ >  0, let3 δ   = min { ǫ/ ( | x 0 | + | y 0 | + 1) , √  ǫ } and note that d ( xy,x 0 y 0 ) ≤| x 0 || y − y 0 | + | y 0 || x − x 0 | + | x − x 0 || y − y 0 | . ](c) Show that the operation of taking reciprocals is a continuous map from  R −{ 0 }  to  R .[Hint: Given  x 0   = 0 and  ǫ >  0, let  δ   = min {| x 0 | / 2 ,x 20 ǫ/ 2 } . Note that  d (1 /x, 1 /x 0 ) = | x − x 0 | / | xx 0 | . If   | x − x 0 |  < δ  , then  | xx 0  − x 20 |  <  | x 0 | 2 / 2, so  xx 0  − x 20  >  − x 20 / 2 and xx 0  > x 20 / 2  >  0.](d) Show that the subtraction and quotient opertions are continuous.2

Sep 22, 2019

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